{
  "id": "2305.17514",
  "version": "v1",
  "published": "2023-05-27T16:21:18.000Z",
  "updated": "2023-05-27T16:21:18.000Z",
  "title": "Some new generalizations of Domination using restrictions on degrees of vertices",
  "authors": [
    "Shyam S. Kamath",
    "Nithya Muraleedharan"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A set $D$ of vertices in a graph $G=(V,E)$ is a degree restricted dominating set for $G$ if each vertex $v_i$ in $D$ is dominating atmost $g(d_i)$ vertices of $V-D$, where $g$ is a function restricting the degree value $d_i$ with respect to the given function value $k_i$ for a natural valued function $f$ from the vertex set of the graph. We define three different types of Degree Restricted Domination by varying the way how the restricted function $g(v_i)$ is defined. If $g(d_i)=\\big\\lceil \\frac{d_i}{k_i}\\big\\rceil$, the corresponding domination is called the ceil degree restricted domination, in short, $CDRD$, and the dominating set obtained in this manner is the $CDRD$-set. If $g(d_i)=\\big\\lfloor\\frac{d_i}{k_i}\\big\\rfloor$ or $g(d_i)=d_i-k_i+1$, then the corresponding dominations are respectively called the floor degree restricted domination, in short $FDRD$, or the translate degree restricted domination, $TDRD$. The dominating sets obtained in this manner are the $FDRD$-set and the $TDRD$-set respectively. In this paper, we introduce these new generalizations of the domination number in line with the different $DRD$-sets and study these types of domination for some classes of graphs like complete graphs, caterpillar graphs etc. Degree restricted domination has a vital role in retaining the efficiency of nodes in a network and has many interesting applications.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-27T16:21:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C07",
      "05C69"
    ],
    "keywords": [
      "generalizations",
      "restrictions",
      "corresponding domination",
      "ceil degree restricted domination",
      "floor degree restricted domination"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}